What is $x$ if $\sin x = x - 2 \pi/3$?
The answer is $x \approx 2.61$ but how do I work that out (without Taylor series - this is homework for 10th grade)? Thanks.
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What tools do you have or you know? :) – Hosein Rahnama Jan 23 '16 at 13:11
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Various simple trigonometric identities (e.g. $\sin 2x = 2 \sin x \cos x$). – Jan 23 '16 at 13:13
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2Then please share your ideas – Archis Welankar Jan 23 '16 at 13:13
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1@ArchisWelankar: I have no idea, which is why I am asking here :D – Jan 23 '16 at 13:15
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Hint use taylor expansion as $x$ is very snall for $sinx$ which is $sin(x)=x-\frac{x^3}{6}+\frac{x^5}{120}+O(x^7)$ and then you get normal polynomial on both sides whose roots can be obtained. Or see the graphs of lhs,rhs where they meet .
Archis Welankar
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Sorry I should have specified that I didn't want to use Taylor series. This was homework for 10th grade. – Jan 23 '16 at 13:06
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As there is no numerical way for you to compute this without using the Taylor Series, I would just solve it graphically. Getting everything on one side, we have that $\sin(x) - x + \frac{2\pi}{3} = 0.$ Let $f(x) = \sin(x) - x + \frac{2\pi}{3},$ plug that into a graphing calculator, and calculate the zero. You find the zero at $\boxed{x = 2.605}.$
K. Jiang
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Yes that's how I got my answer too (actually I just used Wolfram)... but I was hoping for an answer that didn't rely on a graphing calculator. – Jan 24 '16 at 03:35
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Then the only way you could solve this is by using the Taylor Series, which is a calculus topic. – K. Jiang Jan 24 '16 at 17:09